Negative and Positive Temperatures of Static and Dynamic
Aspects

According to definition, total energy
and entropy
are both positive for any system. However, temperature
defined as
may be either positive or negative [1]. Following this definition and the
general criteria [1] for negative temperature, for any well-defined entropies
of any systems of hierarchical structures, correspondingly temperature(s) can
be defined, at least formally. We found that every system has symmetries of
static and dynamic aspects [2] and the two entropies and their
variations can be defined and in principle calculated according to ,
where
is the apparent symmetry number or the order of the group [2].

Therefore, for a conventional thermodynamic system, because information
registration involves reduced static symmetry ()
[2] and the energy increase ()
[3], a negative temperature
(s for static) can be formally defined, while the dynamic motion of such
system has a conventionally understood positive temperature
(d for dynamic).

Similarly, however, for a system of electronic motion in a single atom or a
molecule, the
of the local electronic dynamic motion is found to be negative with the
most negative value at the electronic ground state, while its local
of the static aspect of the electronic structure, such as spin parallel
orientation at excited states, is positive.

It is convenient to use these temperatures to characterize symmetry breaking
phenomena at any one of many hierarchical structures in nature.